If you create a block with entities, then the entities in the block have their own layer and pen attributes. When you make a insert from that block on your drawing, the insert has it own properties (such as layer and pen attributes).
I have tried to change the attributes from the insert (other layer and/or pen attributes), but the insert always displays with the pen attributes from the block. Is this the normal behaviour or am I wrong? Can an insert be made with different properties than the block it is made from?

Try setting the attributes (color / line width / line style) of the entities inside your block to "ByBlock". ByBlock means that the entities will inherit their attributes from the block insert, so you can have instances of the same block with different attributes.

It is not possible to set the layer of an entity to be "ByBlock".

Note that this is very unusual. In almost 100% of all cases you will want to define the entity attributes by layer.